Calculus Examples

$\frac{\frac{x^{2} + 2 x - 3}{x^{2} - x - 6}}{\frac{x - 1}{4 x - 12}}$

Step 1

Multiply the numerator by the reciprocal of the denominator.

$\frac{x^{2} + 2 x - 3}{x^{2} - x - 6} \cdot \frac{4 x - 12}{x - 1}$

Step 2

Factor $x^{2} + 2 x - 3$ using the AC method.

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Step 2.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 3$ and whose sum is $2$ .

$- 1, 3$

Step 2.2

Write the factored form using these integers.

$\frac{(x - 1) (x + 3)}{x^{2} - x - 6} \cdot \frac{4 x - 12}{x - 1}$

Step 3

Factor $x^{2} - x - 6$ using the AC method.

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Step 3.1

Consider the form $x^{2} + b x + c$ . Find a pair of integers whose product is $c$ and whose sum is $b$ . In this case, whose product is $- 6$ and whose sum is $- 1$ .

$- 3, 2$

Step 3.2

Write the factored form using these integers.

$\frac{(x - 1) (x + 3)}{(x - 3) (x + 2)} \cdot \frac{4 x - 12}{x - 1}$

Step 4

Simplify terms.

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Step 4.1

Cancel the common factor of $x - 1$ .

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Step 4.1.1

Cancel the common factor.

$\frac{(x - 1) (x + 3)}{(x - 3) (x + 2)} \cdot \frac{4 x - 12}{x - 1}$

Step 4.1.2

Rewrite the expression.

$\frac{x + 3}{(x - 3) (x + 2)} (4 x - 12)$

Step 4.2

Multiply $\frac{x + 3}{(x - 3) (x + 2)}$ by $4 x - 12$ .

$\frac{(x + 3) (4 x - 12)}{(x - 3) (x + 2)}$

Step 5

Factor $4$ out of $4 x - 12$ .

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Step 5.1

Factor $4$ out of $4 x$ .

$\frac{(x + 3) (4 (x) - 12)}{(x - 3) (x + 2)}$

Step 5.2

Factor $4$ out of $- 12$ .

$\frac{(x + 3) (4 x + 4 \cdot - 3)}{(x - 3) (x + 2)}$

Step 5.3

Factor $4$ out of $4 x + 4 \cdot - 3$ .

$\frac{(x + 3) (4 (x - 3))}{(x - 3) (x + 2)}$

$\frac{(x + 3) \cdot 4 (x - 3)}{(x - 3) (x + 2)}$

Step 6

Cancel the common factor of $x - 3$ .

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Step 6.1

Cancel the common factor.

$\frac{(x + 3) \cdot 4 (x - 3)}{(x - 3) (x + 2)}$

Step 6.2

Rewrite the expression.

$\frac{(x + 3) \cdot 4}{x + 2}$

Step 7

Move $4$ to the left of $x + 3$ .

$\frac{4 (x + 3)}{x + 2}$